Pythagorean Theorem

Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Formally: If a right triangle has legs of lengths $a$ and $b$, and hypotenuse of length $c$, then:

$$a^2 + b^2 = c^2$$

To understand this proof, you should be familiar with:

• Basic properties of triangles
• Similar triangles
• Basic algebraic manipulations

The Pythagorean Theorem describes a fundamental relationship in right triangles. Geometrically, it states that the area of the square built on the hypotenuse equals the sum of areas of squares built on the legs. This relationship is central to Euclidean geometry and has numerous applications in mathematics, physics, engineering, and architecture.

We will demonstrate a proof based on similar triangles, attributed to Euclid.

Main Proof:

Consider a right triangle ABC with right angle at C. Draw the altitude from C to the hypotenuse AB, creating point D on AB.

Step 1: Establish similar triangles.

The triangles ABC, ADC, and BDC are all similar to each other. This is because: - Triangle ABC has a right angle at C - Triangle ADC has a right angle at C - Triangle BDC has a right angle at C - They all share various angles, making them similar by the Angle-Angle similarity criterion

Step 2: Use properties of similar triangles.

Since triangles ADC and ABC are similar: $$\frac{AD}{AB} = \frac{AC}{AB} = \frac{AC}{c}$$

Similarly, since triangles BDC and ABC are similar: $$\frac{BD}{AB} = \frac{BC}{AB} = \frac{BC}{c}$$

Step 3: Express lengths in terms of the altitude.

Let's call the length of the altitude from C to AB as $h$. Since triangles ADC and BDC both have right angles:

$$AC^2 = AD \cdot AB = AD \cdot c$$ $$BC^2 = BD \cdot AB = BD \cdot c$$

Step 4: Add the equations.

$$AC^2 + BC^2 = AD \cdot c + BD \cdot c = (AD + BD) \cdot c = AB \cdot c = c \cdot c = c^2$$

Since $AC = a$ and $BC = b$, we have:

$$a^2 + b^2 = c^2$$

Therefore, the square of the hypotenuse equals the sum of the squares of the legs. ■

Alternative Proof 1: Using Area Decomposition

Another elegant proof uses the decomposition of squares:

Area-based proof of the Pythagorean Theorem

Alternative Proof 2: Algebraic Proof

A more algebraic approach uses coordinate geometry:

Algebraic proof of the Pythagorean Theorem

The Pythagorean Theorem has numerous applications:

1. Distance in Cartesian coordinates: The formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ is derived directly from the Pythagorean Theorem.

2. Trigonometry: The theorem provides the foundation for defining trigonometric functions and establishing trigonometric identities.

3. Navigation and Surveying: Used to calculate distances and positions.

4. Physics: Applied in vector decomposition and calculation of resultant forces.

While commonly attributed to Pythagoras (c. 570–495 BCE), evidence suggests this relationship was known earlier in Babylonian, Egyptian, and Chinese mathematics.

Pythagoras or his followers may have been the first to provide a formal mathematical proof of the theorem. The oldest recorded statement and proof appear in Euclid's “Elements” (c. 300 BCE).

Various cultures have discovered and proven this theorem independently, demonstrating its fundamental nature in mathematics.

• [1] Euclid. “Elements,” Book I, Proposition 47.
• [2] Heath, T. L. (1956). “The Thirteen Books of Euclid's Elements.” Dover Publications.
• [3] Maor, E. (2007). “The Pythagorean Theorem: A 4,000-Year History.” Princeton University Press.
• [4] Loomis, E. S. (1968). “The Pythagorean Proposition.” National Council of Teachers of Mathematics.

• Law of Cosines - A generalization of the Pythagorean Theorem to non-right triangles
• Distance Formula in Cartesian Coordinates
• Converse of the Pythagorean Theorem